Convex hulls of surfaces in fourspace

TL;DR

This paper investigates the algebraic boundaries of convex hulls of algebraic surfaces in four-dimensional space, revealing unique geometric phenomena and analyzing specific surface classes using advanced algebraic methods.

Contribution

It provides a detailed analysis of the algebraic boundary of convex hulls of surfaces in fourspace, highlighting new geometric phenomena not seen in lower dimensions.

Findings

Identification of new geometric phenomena in fourspace surfaces

Analysis of algebraic boundaries of Veronese, Del Pezzo, and Bordiga surfaces

Application of Ranestad and Sturmfels' formula to real algebraic surfaces

Abstract

This is a case study of the algebraic boundary of convex hulls of varieties. We focus on surfaces in fourspace to showcase new geometric phenomena that neither curves nor hypersurfaces do. Our method is a detailed analysis of a general purpose formula by Ranestad and Sturmfels in the case of smooth real algebraic surfaces of low degree (that are rational over the complex numbers). We study both the complex and the real features of the algebraic boundary of Veronese, Del Pezzo and Bordiga surfaces.